Distortion Risk Measures
and Economic Capital
Discussion of
Werner Hurlimann Paper --By Shaun Wang
April 11, 2003
1
Agenda

Highlights of W. Hurlimann Paper:
Search for distortion measures that preserve
an order of tail heaviness
2) Optimal level of capital
1)

Discussion by S. Wang:
Link distortion measures to financial pricing
theories
2) Empirical studies in Cat-bond, corporate bond
1)
April 11, 2003
2
Assumptions

We know the dist’n F(x) for financial losses


In real-life this may be the hardest part
Risks are compared solely based on F(x)

Correlation implicitly reflected in the aggregate
risk distribution
April 11, 2003
3
Axioms for Coherent Measures

Axiom 1. If X  Y  (X)  (Y).

Axiom 2. (X+Y)  (X)+ (Y)


Axiom 3. X and Y are co-monotone
 (X+Y) = (X)+ (Y)
Axiom 4.
April 11, 2003
Continuity
4
Representation for Coherent
Measures of Risk

Given Axioms 1-4, there is distortion
g:[0,1][0,1] increasing concave with
g(0)=0 and g(1)=1, such that
F*(x) = g[F(x)] and (X) = E*[X]

Alternatively, S*(x) = h[S(x)], with
S(x)=1F(x) and h(u)=1 g(1u)
April 11, 2003
5
Some Coherent Distortions

TVaR or CTE: g(u) = max{0, (u)/(1)}

PH-transform: S*(x) = [S(x)]^, for  <1

Wang transform:
g(u) = [1(u)+],
where  is the Normal(0,1) distribution
April 11, 2003
6
Distortion Risk Measures
Distortion Functions
F*(x)=g[F(x)]
1
PH-transform
0.8
TVaR
F*(x)
0.6
Wang transform
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
F(x)
April 11, 2003
7
Distortion Risk Measures
Transformed Probability Weights
0.12
0.1
f*(x)
0.08
PH-transform
TVaR
0.06
Wang transform
0.04
f(x)
0.02
0
0
20
40
60
80
100
f(x) --- Uniform(0, 100) Density
April 11, 2003
8
Ordering of Tail Heaviness



Hurlimann compares risks X and Y with
equal mean and equal variance
If E[(X c)+^2]  E[(Y c)+^2] for all c, Y
has a heavier tail than risk X
He tries to find “distortion measures” that
preserve his order of tail heaviness
April 11, 2003
9
Hurlimann Result

For the families of bi-atomic risks and 3parameter Pareto risks,

A specific PH-transform: S*(x)=[S(x)]0.5
preserves his order of tail heaviness

Wang transform and TVaR do not
preserve his order of tail thickness
April 11, 2003
10
Optimal Risk Capital
Definitions :
 Economic Risk Capital: Amount of capital
required as cushion against potential
unexpected losses
 Cost of capital: Interest cost of financing

Excess return over risk-free rate demanded by
investors
April 11, 2003
11
Optimal Risk Capital: Notations




X:
financial loss in 1-year
C = C[X]:
economic risk capital
i
borrowing interest rate
r< i
risk-free interest rate
April 11, 2003
12
Dilemma of Capital Requirement

Net interest on capital (i  r)C  small C

Solvency risk

Let R[.] be a risk measure to price insolvency


X  C(1+r)  large C
See guarantee fund premium by David Cummins
Minimize total cost:
R[max{X  C(1+r),0}] + (i  r)C
April 11, 2003
13
Optimal Risk Capital: Result

Optimal Capital (Dhane and Goovaerts, 2002):
C[X] = VaR(X)/(1+r)

 =1 g1[(i  r)/(1+r)]



with
When (i  r) increases, optimal capital
decreases!
Eg. XNormal(,), i=7.5%, r=3.75%, and
g(u)=u^0.5,  C[X]=[+3]/1.0375
April 11, 2003
14
Remarks



In standalone risk evaluation, distortion
measures may or may not preserve
Hurlimann’s order of tail heaviness
However, individual risk distribution tails
can shrink within portfolio diversification
We need to reflect the portfolio effect and
link with financial pricing theories
April 11, 2003
15
Properties of Wang transform

F * ( x )    ( F ( x ))  

1

If the asset return R has a normal distribution
F(x), transformed F*(x) is also normal with


E*[R] = E[R]   [R] = r (risk-free rate)
 = { E[R] r }/[R] is the “market price of risk”,
also called the Sharpe ratio
April 11, 2003
16
Link to Financial Theories



Market portfolio Z has market price of risk 0
corr(X,Z) = 
Buhlmann 1980 economic model 


F * ( x )    1 ( F ( x ))   , with   0
• It recovers CAPM for assets, and BlackScholes formula for Options
April 11, 2003
17
Unified Treatment of Asset /
Loss


The gain X for one party is the loss for the
counter party: Y = X
We should use opposite signs of , and we get
the same price for both sides of the transaction
FX* ( x )    1 ( FX ( x ))   
FY* ( x )    1 ( FY ( x ))   
April 11, 2003
18
Risk Adjustment for Long-Tailed
Liabilities

The Sharpe Ratio  can adjust for the time
horizon:
(T) = (1) * (T)b, where 0.5 b 1
 where
T is the average duration of loss payout
patterns
 b=0.5 if reserve development follows a
Brownian motion
April 11, 2003
19
Adjustment for Parameter
Uncertainty
From Normal to Student-t
0.05
Normal(0,1)
0.04
prob density
Student(k=5)
0.03
0.02
0.01
0
X
April 11, 2003
20
Adjust for Parameter
Uncertainty

Baseline: For normal distributions, Student-t
properly reflects the parameter uncertainty

Generalization: For arbitrary F(x), we propose
the following adjustment:

F(x)  Normal(0,1)   Student-t Q
F ( x )  Q  ( F ( x )) 
*
April 11, 2003
1
21
A Two-Factor Model

First adjust for parameter uncertainty


F(x)  Normal(0,1)   Student-t Q
Then Apply Wang transform:
F * ( y )  Q  ( F ( y ))   
1
April 11, 2003
22
Fit 2-factor model to 1999 Cat bonds
Date Sources: Lane Financial LLC
Yield Spread for Insurance-Linked Securities
16.00%
Yield Spread
14.00%
Model-Spread
Empirical-Spread
12.00%
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
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Transactions
April 11, 2003
23
Fit 2-factor model to corporate bonds
Bond Rating and Yield Spread
1,400
Model Fitted Spread
1,200
Spread (basis points)
Actual Spread
1,000
800
600
400
200
0
AAA
AA
A
BBB
BB
B
CCC
Bond Rating
April 11, 2003
24